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The de Broglie-Schr\"odinger Trajectories

A. Orefice, Raffaele Giovanelli, Domenico Ditto

Published Mar 4, 2019• 0 citations
Publisher

Abstract

De Broglie's quest for a wave-like approach capable of representing the position of moving particles is satisfied by assuming that they run along the trajectories associated, under assigned starting conditions, with a time-independent Schrodinger (or Klein-Gordon) equation. Just like in usual Dynamics, the starting conditions determine a set of virtual trajectories independent from the number, and even from the very presence, of moving particles.

DOI
10.13140/rg.2.2.20013.44002/1

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De Broglie's quest for a wave-like approach capable of representing the position of moving particles is satisfied by assuming that they run along the trajectories associated, under assigned starting conditions, with a time-independent Schrodinger (or Klein-Gordon) equation. Just like in usual Dynamics, the starting conditions determine a set of virtual trajectories independent from the number, and even from the very presence, of moving particles.
De Broglie's quest for a wave-like approach capable of representing the position of a moving particle, is satisfied, in the case of time-independent external fields, by assuming that each particle runs along the virtual trajectories associated, under assigned starting conditions, with a time-independent Schrödinger (or Klein-Gordon) equation. Just like in classical Dynamics, indeed, the starting conditions determine, from the beginning, a set of virtual particle trajectories independent from the very presence of the particles.
The time-independent Schroedinger and Klein-Gordon equations - as well as any other Helmholtz-like equation - were recently shown to be associated with exact sets of ray-trajectories (coupled by a "Wave Potential" function encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory.
An exact, ray-based general treatment is shown to hold for any kind of monochromatic wave feature - including diffraction and interference - described by Helmholtz-like equations, under the coupling action of a dispersive function (which we call "Wave Potential") encoded in the structure itself of the Helmholtz equation. Since the time-independent Schroedinger and Klein-Gordon equations (associating particles of assigned total energy with stationary de Broglie waves) are themselves Helmholtz-like equations, the same general approach is extended to the ray-based dynamics of point-like particles, whose exact trajectories and dynamical laws may be obtained without resorting to statistical concepts of any kind, thus suggesting a non-probabilistic nature of de Broglie's waves and of physical reality.
We illustrate, using a simple model, that in the usual formulation the time-component of the Klein-Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if one uses a covariant extension of the canonical formulation of relativistic point particle dynamics proposed by Gitman and Tyutin.
The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.